Advanced Research Methods II
Advanced Research Methods II SW 983
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This 4 page Class Notes was uploaded by Mr. Joseph Beer on Monday September 7, 2015. The Class Notes belongs to SW 983 at Kansas taught by Tom McDonald in Fall. Since its upload, it has received 25 views. For similar materials see /class/182414/sw-983-kansas in Social Welfare at Kansas.
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Date Created: 09/07/15
SW 983 LECTURE NOTES FACTOR ANALYSIS Uses of Factor Analysis Data Reduction Determining underlying structures These two are not mutually exclusive In carrying out 1 it is necessary to assumediscover 2 If one starts with 2 you should have some hypotheses in mind Exploratory FA searching for underlying structure Confirmatory FA testing hypotheses about underlying structure which generated set of measures Factor F is a construct latent variable or unmeasured variable X s are indicators or measured variables Conducting Factor Analysis 1 Examine the correlation matrix since one of the goals of factor analysis is to obtain quotfactorsquot that help explain the correlations between variables the variables must be related to each other for the factor model to be appropriate Bartlett39s test for sphericity can be used to test the hypothesis that the correlation matrix is an identity matrix If there is an underlying structure the variables should be correlated You wish to reject the null hypothesis if the data are suitable for FA KaiserMeyerOlkin measure of sampling adequacy referred to as MSA in our book is an index for comparing the magnitudes of the observed correlation coefficeints to the magnitude of the partial correlation coefficients partial correlations should be small if the variables share common factors Values closer to l gt 5 needed for factor analysis 2 Factor extraction Principle components and others the first principal component is the combination that explains the largest amount of variance in the sample Successive components explain progressively less This is less true for principle axis factoring making it more appropriate if we are looking for structure not just data reduction Communality common factor variance sums of squares of factor loadings the proportion of variability for a given variable that is explained by the factors Factor Loading correlations between variables and factors if orthogonal coefficients used to express a standardized variable in terms of the factors Factor Matrix table of factor loadings Eigen Values variance explained by each factor using standardized variables Cutoff generally at values gt 1 Scree Plot 7 plot of total variance associated with each factor Scree begins at last true factor Factor Structure Matrix 7 contains correlations between variables and factors Same as pattern matrix when factors are orthogonal Factor Pattern Matrix 7 contains factor loadings regression of variables on factors 3 Factor rotation unrotated matrix frequently not interpretable The goal of rotation is to transform complicated matrices into simpler onesl Obligue Rotation 7 factors are allowed to be correlated Generally this is a reasonable assumption with the types of variable lists we will be dealing with You can run oblique first then check the correlations of the resulting factors If small try orthogonal rotation Orthogonal Rotation factors not correlated o Varimax minimizes number of variables that have high loadings on a factor thereby enhancing interpretability Most commonly used 0 Quartimax minimizes the number of factors needed to explain a variable often results in a general factor with high to moderate loadings on most variables 0 Eguamax combination of Varimax and Quartimax 4 Interpret and label factors With orthogonal rotation the pattern and structure matrices are identical With oblique rotation they are different The pattern matrix is best for determining the clusters of variables defined by the oblique factors 5 Calculate factor scores ij Z wjixik where i number of variables all variables are included j number of factors k number of cases i or create summated scale 2 x 139 where only the variables with significant loadings are summed iNOTE OnTotalVarianceExplainedTable First set of variances always from PC regardless of the extraction method chosen Second set will differ if PFA or anything other than PC used as extraction method Orthogonal rotation yields third set where accounted for by each factor will differ but cumulative will be the same as extraction When factors are correlated sums of squares loadings cannot be added to obtain total variance SW983 CODING OF CATEGORICAL VARIABLES IN MULTIPLE REGRESSION ANALYSIS Overview Multiple regression analysis MR and other correlation based statistical techniques like factor analysis are typically used for continuous variables It was developed as an extension of correlation partial and semi partial The techniques discussed today broaden its application to include the use of categorical variables The goal then is to learn how categorical variables can be introduced to MR analysis and how the results can be interpreted Recall that categorical variables can be used to measure group membership Subjects differ from each other in type or kind rather than in degree continuous The categorical or discrete variable may re ect assignment to a group eg experimental or control group or some attribute variable e g sex of respondent Note Methods for coding categorical variables are the same regardless of whether the data are experimental or nonexperimental explanatoy or predictive Codi g Coding of categorical variables is not simply a statistical or mathematical or clerical function but depends upon the validity of underlying conceptualizations to be useful eg treatment vs control It is important to remember that what we are doing throughout this chapter is comparing means across groups just as we did using ttests and AN OVA The overall results will always be the same regardless of the coding method used dummy effect and orthogonal and these results will be identical to ttest or ANOVA when more than two groups are involved Dummy Coding 1 most common Definition Use of zero39s and one s to denote group membership called indicator coding in our text 0 not a member of group 1 member of group Rule Number of dummy variables needed k Number of groups g l Dummy coding allows multiple regression to be used to compare group means ie metric dependent variable and two groups Results will be identical to ttests in the case of just two groups or oneway ANOVA in the case of more than two groups Consider the equation Y b0 Z kak k Where Y39predicted value for any individual mean for the group b0 mean for the omitted group bk difference in mean for represented group and omitted group T k F 0 Xk 1 if member of kth group other wise 0 SSreg SSbetween groups SSresidual SSwithin groups R2 Eta2 ANOVA F t2 two groups F ANOVA more than two groups b0 intercept mean for omitted group b1 difference between group 1 and omitted group b2 difference between group 2 and omitted group etc t ratio statistic for each b coefficient is a test of the signi cance of the difference between means of the omitted group and the group represented by the particular dummy variable associated with b Note that this is not available from oneway unless you run contrasts Effect Coding Fixed Effects Linear Model Definition Like dummy coding except that the omitted group is assigned l39s rather than 0 s in all the vectors ie group variables Remember overall results will not differ with effect or dummy coding However interpretation of the regression coefficients is very different Regression coefficients not re ect the effects of treatments ie deviations from the grand mean b0 F bk deviation of mean for group i from grand mean Y39 mean for the group To calculate the b for the omitted group remember that Z bg 0
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